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Theorem brab2ga 5064
Description: The law of concretion for a binary relation. See brab2a 5038 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
brab2ga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brab2ga.2  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) }
Assertion
Ref Expression
brab2ga  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)

Proof of Theorem brab2ga
StepHypRef Expression
1 brab2ga.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) }
2 opabssxp 5063 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } 
C_  ( C  X.  D )
31, 2eqsstri 3519 . . 3  |-  R  C_  ( C  X.  D
)
43brel 5037 . 2  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)
5 df-br 4440 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
61eleq2i 2532 . . . 4  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
75, 6bitri 249 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
8 brab2ga.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
98opelopab2a 4751 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
107, 9syl5bb 257 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ps ) )
114, 10biadan2 640 1  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022   class class class wbr 4439   {copab 4496    X. cxp 4986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994
This theorem is referenced by:  fnse  6890  ltxrlt  9644  ltxr  11327  gaorb  16544  ispgp  16811  efgcpbllema  16971  lmbr  19926  isphtpc  21660  vitalilem1  22183  vitalilem2  22184  vitalilem3  22185  filnetlem1  30436
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